What you'll learn
This revision guide covers how to identify and calculate resultant forces acting on objects, and how to represent these forces using free body diagrams. You'll learn to resolve forces in different directions, understand equilibrium conditions, and apply Newton's laws to predict object motion. These skills are essential for many calculations in GCSE Physics and frequently appear in exam questions worth 4-6 marks.
Key terms and definitions
Force — a push or pull acting on an object, measured in newtons (N), that can change the object's velocity, direction or shape.
Resultant force — the single force that has the same effect as all the original forces acting on an object combined together; also called the net force.
Free body diagram — a simplified diagram showing an object as a point or box with arrows representing all the forces acting on it, with arrow length proportional to force magnitude.
Equilibrium — the state of an object when all forces acting on it are balanced, producing a resultant force of zero, meaning the object remains at rest or moves at constant velocity.
Vector quantity — a quantity with both magnitude (size) and direction, such as force, velocity or displacement.
Component — one part of a force when resolved into perpendicular directions, typically horizontal and vertical.
Contact force — a force that acts when objects are physically touching, such as friction, air resistance, tension or normal contact force.
Non-contact force — a force that acts at a distance without physical contact, such as gravitational force, electrostatic force or magnetic force.
Core concepts
Understanding forces as vectors
Forces are vector quantities, which means they have both magnitude and direction. When representing forces, you must show:
- The magnitude using a number with the unit newtons (N)
- The direction using an arrow or compass direction (north, south, east, west) or angles
- For calculations, forces in opposite directions have opposite signs (e.g., upward forces are positive, downward forces are negative)
Multiple forces can act on an object simultaneously. The resultant force determines what happens to the object's motion according to Newton's laws. If forces act along the same straight line, you can add or subtract them directly. If forces act in different directions, you need to use vector addition or resolve them into components.
Drawing accurate free body diagrams
A free body diagram is a fundamental tool for analysing forces. Follow these steps to draw one correctly:
Step 1: Draw the object as a simple shape (usually a box or dot) to represent its centre of mass
Step 2: Identify all forces acting ON the object (not forces the object exerts on other things)
Step 3: Draw each force as an arrow starting from the object:
- The arrow points in the direction the force acts
- The arrow length represents the force magnitude (use a scale if values are given)
- Label each arrow clearly with the force name and value if known
Step 4: Check you've included all relevant forces for the situation
Common forces to consider include:
- Weight (W or mg) — always acts vertically downward from the centre of mass
- Normal contact force (N or R) — acts perpendicular to the surface, pushing the object away from it
- Friction (F) — acts parallel to surfaces, opposing motion or attempted motion
- Air resistance or drag (D) — opposes motion through air or fluid
- Tension (T) — acts along strings, ropes or cables, pulling on the object
- Thrust or driving force — pushes or pulls the object forward
Calculating resultant forces in one dimension
When all forces act along a single straight line (one-dimensional motion), calculate the resultant force by adding forces in one direction and subtracting forces in the opposite direction.
Method:
- Choose a positive direction (usually right or upward)
- Add all forces acting in the positive direction
- Subtract all forces acting in the negative direction
- The result is the resultant force with its direction
Example: A car experiences a driving force of 1200 N forward and friction of 300 N backward.
Resultant force = 1200 N - 300 N = 900 N forward
If the resultant force is:
- Positive: the object accelerates in the positive direction
- Negative: the object accelerates in the negative direction
- Zero: the object is in equilibrium (stationary or constant velocity)
Calculating resultant forces in two dimensions
When forces act in different directions (not along the same line), you need to find the resultant using vector addition. For perpendicular forces, use Pythagoras' theorem and trigonometry.
For two perpendicular forces:
Magnitude of resultant force: $$F_R = \sqrt{F_1^2 + F_2^2}$$
Direction (angle θ from horizontal force): $$\tan θ = \frac{F_{vertical}}{F_{horizontal}}$$
Example: A force of 30 N acts horizontally, and a force of 40 N acts vertically on an object.
Resultant force magnitude: $$F_R = \sqrt{30^2 + 40^2} = \sqrt{900 + 1600} = \sqrt{2500} = 50 \text{ N}$$
Direction: $$\tan θ = \frac{40}{30} = 1.33$$ $$θ = \tan^{-1}(1.33) = 53°$$ above the horizontal
Resolving forces into components
Sometimes you need to break a single force into perpendicular components (usually horizontal and vertical). This is called resolving a force.
For a force F at angle θ to the horizontal:
- Horizontal component: $F_x = F \cos θ$
- Vertical component: $F_y = F \sin θ$
This technique is useful when:
- An object moves on a slope
- Forces act at angles
- You need to analyse motion in perpendicular directions separately
Example: A force of 100 N acts at 30° to the horizontal.
Horizontal component = 100 × cos 30° = 100 × 0.866 = 86.6 N
Vertical component = 100 × sin 30° = 100 × 0.5 = 50 N
Equilibrium and balanced forces
An object is in equilibrium when the resultant force is zero. This occurs when:
- All horizontal forces balance (sum to zero)
- All vertical forces balance (sum to zero)
Objects in equilibrium either:
- Remain at rest (stationary)
- Continue moving at constant velocity in a straight line
This is Newton's First Law of Motion in action. No resultant force means no acceleration.
For an object in equilibrium:
- Upward forces = Downward forces
- Forward forces = Backward forces
Example: A book resting on a table experiences weight (10 N downward) and normal contact force (10 N upward). Resultant force = 0 N, so the book remains stationary.
Worked examples
Example 1: Free body diagram and resultant force
Question: A box of mass 5 kg is pushed across a horizontal floor with a force of 30 N. Friction opposes the motion with a force of 12 N. Draw a free body diagram for the box and calculate the resultant force. [4 marks]
Solution:
Free body diagram should show: [2 marks]
- Box drawn as a square or rectangle
- Weight (49 N or 50 N) acting downward
- Normal contact force (49 N or 50 N) acting upward
- Applied force (30 N) acting horizontally to the right
- Friction (12 N) acting horizontally to the left
- All forces correctly labelled with values and directions
Calculation: [2 marks]
Weight = mass × gravitational field strength = 5 kg × 9.8 N/kg ≈ 50 N [Accept 49 N]
Vertical forces: 50 N up - 50 N down = 0 N (balanced) [1 mark]
Horizontal forces: 30 N right - 12 N left = 18 N to the right [1 mark]
Resultant force = 18 N to the right (or in the direction of motion)
Example 2: Two-dimensional resultant force
Question: A sailing boat experiences a forward thrust of 800 N from the wind and a current pushing it sideways with a force of 600 N. Calculate the magnitude and direction of the resultant force on the boat. [4 marks]
Solution:
The two forces are perpendicular to each other. [1 mark for stating or showing this]
Using Pythagoras' theorem: $$F_R = \sqrt{800^2 + 600^2}$$ [1 mark]
$$F_R = \sqrt{640000 + 360000} = \sqrt{1000000}$$
$$F_R = 1000 \text{ N}$$ [1 mark]
Using trigonometry to find direction: $$\tan θ = \frac{600}{800} = 0.75$$
$$θ = \tan^{-1}(0.75) = 36.9°$$ or 37° [1 mark]
The resultant force is 1000 N at 37° to the forward direction (toward the side where the current pushes).
Example 3: Equilibrium on a slope
Question: A car of mass 1200 kg is parked on a hill. The component of weight acting down the slope is 2400 N. The brakes provide a friction force up the slope. Calculate the friction force needed to keep the car in equilibrium. [3 marks]
Solution:
For the car to be in equilibrium, the resultant force must be zero. [1 mark]
Forces parallel to the slope must balance. [1 mark]
Friction force up slope = Component of weight down slope
Friction force = 2400 N up the slope [1 mark]
Common mistakes and how to avoid them
Including forces the object exerts on other things — Free body diagrams show only forces acting ON the object, not forces FROM the object. For example, if drawing forces on a book on a table, don't include the force the book exerts on the table.
Drawing weight from the wrong point — Weight always acts from the centre of mass (usually the centre of the object), not from the base or point of contact with a surface.
Forgetting to include all forces — Systematically check for weight, normal contact force, friction/air resistance, and any applied forces. Weight acts on all objects near Earth's surface.
Incorrect arrow lengths in diagrams — Arrow length must represent force magnitude. If one force is twice another, its arrow should be twice as long. Use a ruler and scale.
Sign errors in calculations — Be consistent with your positive direction. If up is positive, all downward forces must be negative. If right is positive, all leftward forces must be negative.
Using the wrong trigonometry function — Remember SOH CAH TOA: sine for opposite/hypotenuse, cosine for adjacent/hypotenuse, tangent for opposite/adjacent. Check your calculator is in degree mode.
Exam technique for "Resultant forces and free body diagrams"
Command word "Draw" — For free body diagrams, use a ruler for straight arrows, label all forces clearly, and ensure arrow lengths reflect relative magnitudes. You'll typically get 2-3 marks: 1 for the object representation, 1-2 for correct forces with labels.
Command word "Calculate" — Show all working clearly. Write the formula, substitute values with units, and give the answer with the correct unit. Most calculation questions award 1 mark for method and 1 mark for correct answer. State the direction for vector quantities.
Multi-step problems — Questions often combine free body diagrams with calculations. Draw the diagram first to visualise all forces, then use it to identify which forces to include in your calculation. Check vertical and horizontal forces separately.
Mark allocation guide — 1 mark questions usually need a simple statement or single calculation step. 3-4 mark questions require a free body diagram or multi-step calculation with clear working. 5-6 mark questions combine diagrams, calculations and explanations.
Quick revision summary
Forces are vectors with magnitude and direction measured in newtons. The resultant force is the combined effect of all forces acting on an object. Free body diagrams show all forces acting on an object using labelled arrows from a central point. For one-dimensional motion, add forces in one direction and subtract opposing forces. For perpendicular forces, use Pythagoras' theorem and trigonometry. Objects are in equilibrium when the resultant force is zero, meaning they stay at rest or move at constant velocity. Always show working in calculations and draw diagrams with a ruler.